NLA editor, I can't get two baked animations to work at the same time
In your NLA-Editor's properties panel (on the right); if you select your "movement" action, you can see the Blending mode. In the screenshot you provided it is set to Replace. Replace tells blender that any other tracks' keys on the same channel that are below it are overridden by the top track. You can set this to another type, for instance Add to add the location of the movement-action to the values in the idle-action.

A better way (in my opinion) would be to pay attention that the channels of your actions do not unnecessarily overlap. That means that your walk-cycle should not have keys for body-movement, and your movement-action has only keys for body-movement. If you have that you will have more flexibility in arranging the tracks later on. (In case that you design your walk-cycle by selecting all bones of your armature, and then do "insert key frame", then you will end up with a key in every possible channel, making it very hard to handle in the NLA-Editor. Better remove the unnecessary channel in the action editor then). For example, for a walk-cycle, i would only animate the up/down movements of an in-place walk cycle in one action, and make a different action for movement that effects only the x-direction of the root-bone of my armature, but nothing else.

Baking is mostly not needed for animations, as long as you have enough channels (and bones) to realize your animation. Sometimes, however, when a rig is too simple to support overlaying actions well, it might be necessary.